Reduced-Complexity Power-Efficient Wireless
OFDMA using an Equally Probable CSI Quantizer
Antonio G. Marques∗ , Fadel F. Digham† , Georgios B. Giannakis (contact author)† , and F. Javier Ramos∗
∗ Dept.
of Signal Theory and Communications. Universidad Rey Juan Carlos. Fuenlabrada, 28943 Madrid (SPAIN)
Email: {antonio.garcia.marques, javier.ramos}@urjc.es
† Dept. of Electrical and Computer Engineering. University of Minnesota. Minneapolis, 55455 MN (USA)
Email: {georgios, fdigham}@ece.umn.edu
Abstract— Emerging applications involving low-cost wireless
sensor networks motivate well optimization of multi-user orthogonal frequency-division multiple access (OFDMA) in the powerlimited regime. In this context, the present paper relies on limitedrate feedback (LRF) sent from the access point to terminals
to acquire quantized channel state information (CSI) in order
to minimize the total average transmit-power under individual
average rate and error probability constraints. Specifically, we
introduce two suboptimal reduced-complexity schemes to: (i)
allocate power, rate and subcarriers across users; and (ii) design
accordingly the channel quantizer. The latter relies on the
solution of (i) to design equally probable quantization regions
per subcarrier and user. Numerical examples corroborate the
analytical claims and reveal that the power savings achieved by
our reduced-complexity LRF designs are close to those achieved
by the optimal solution.
I. I NTRODUCTION
Orthogonal frequency-division multiplexing (OFDM) is the
most common modulation for bandwidth limited wireline
and wireless transmissions over frequency-selective multipath
channels. OFDM transmissions over wireline or slowly fading
wireless links have traditionally relied on deterministic or perfect (P-) channel state information at the transmitters (CSIT)
to adaptively load power, bits and/or subcarriers so as to either
maximize rate (capacity) for a prescribed transmit-power, or,
minimize power subject to instantaneous rate constraints [9].
While the assumptions of P-CSI at the transmitters and
receiver render analysis and design tractable, they may not be
as realistic due to wireless channel variations and estimation
errors, feedback delay, bandwidth limitation, and jamming
induced errors [6]. These considerations motivate a limitedrate feedback (LRF) mode, where only quantized (Q-) CSIT
is available through a (typically small) number of bits fed
back from the receiver to the transmitters; see e.g., [10]. QCSIT entails a finite number of quantization regions describing different clusters of channel realizations [7], [10]. Upon
estimating the channel, the receiver feeds back the index of
the region individual uplink channels belong to (channel codeword), based on which each terminal adapts its transmission
parameters accordingly. This LRF-based mode of operation
Work in this paper was supported by the US ARL under the CTA Program,
Cooperative Agreement No. DAAD19-01-2-0011; by the USDoD ARO grant
No. W911NF-05-1-0283; and by the Government of C.A. Madrid under grant
No. P-TIC-000223-0505.
fulfills two requirements: (i) the feedback is pragmatically
affordable in most practical wireless links, and (ii) the Q-CIST
is robust to channel uncertainties since transmitters adapt to a
few regions rather than individual channel realizations.
Resource allocation in orthogonal frequency-division multiple access (OFDMA) minimizing the transmit-power per
symbol based on P-CSIT was first studied in [9]. Relying
on fixed (as opposed to adaptive) Q-CSIT, recent works deal
with optimization of power or rate performance per OFDMA
symbol [2], [5]. Different from these works, here we jointly
adapt power, rate, and subcarrier resources based on Q-CSIT
to minimize the average transmit-power. Our focus is on
allocation algorithms with negligible on-line computational
complexity. Moreover, we rely on the optimal allocation for
designing a novel non-iterative channel quantizer that enforces
equally probable quantization regions per user and subcarrier.
The rest of the paper is organized as follows. After introducing preliminaries on the setup we deal with (Section II),
for a given quantizer design, we derive suboptimal subcarrier,
power, and bit OFDMA allocation (Section III). Once the
allocation is characterized, we capitalize on it for designing
a quantizer with equally probable regions (Section IV). Numerical results and comparisons that corroborate our claims
are presented (Section V), and concluding remarks finish this
paper (Section VI)1 .
II. P RELIMINARIES AND P ROBLEM S TATEMENT
We consider a wireless OFDMA system (see Fig. 1) with
M users, indexed by m ∈ [1, M ], sharing K subcarriers
(subchannels), indexed by k ∈ [1, K]. The instantaneous (per
symbol) power and rate user m loads on subcarrier k are
denoted by pk,m and rk,m , respectively. With these as entries
we form K × M instantaneous power and rate matrices P and
R, that is [P]k,m := pk,m and [R]k,m := rk,m . For a given
1 Notation: Lower and upper case boldface letters are used to denote
(column) vectors and matrices, respectively; (·)T denotes transpose; [·]k,l
the (k, l)th entry of a matrix, and [·]k the kth entry of a vector; X ≥ 0
means all entries of X are nonnegative; FN stands for the normalized FFT
2π
matrix with entries [FN ]n,k = e−j N kn , n, k = 0, . . . , N − 1; fX (X)
denotes the joint probability density function (PDF) of a matrix X; likewise,
fx (x) denotes the PDF of a scalar x; EX [·] stands for the expectation operator
over X; b·c (d·e) denotes the floor (ceiling) operation; I{·} is short for the
indicator function; i.e., I{x} = 1 if x is true and zero otherwise; and LHS(x)
denotes the left hand side of equation (x).
Fig. 1.
System block diagram.
feedback update, we consider a time sharing user access per
subcarrier; i.e., time division multiple access (TDMA)2 . This
sharing process is described by the K × M weight matrix
W whose (k, m)th entry wk,m represents the percentage of
time the kth subcarrier is utilized by the mth user. Clearly,
P
M
m=1 wk,m ≤ 1, ∀k, and the average power and rate over
the transmission period between successive feedback updates
is pk,m wk,m and rk,m wk,m for the kth subcarrier of user m.
Each user’s discrete-time baseband equivalent impulse
response of the corresponding frequency-selective fading
channel is hm := [hm,0 , . . . , hm,Nm ]T , where: Nm :=
bDm,max /Ts c denotes the channel order, Dm,max the maximum delay spread, Ts the sampling period, and Nmax :=
maxm∈[1,M ] Nm,max . As usual in OFDM, we suppose K À
Nmax . For notational convenience, we collect the M impulse
response vectors in a K×M matrix H := [h1 , . . . , hM ], where
the length of each column is increased to K by padding an
appropriate number of zeros.
Each user applies a K-point inverse fast Fourier transform
(I-FFT) to each snapshot of K-symbol streams, and subsequently inserts a cyclic prefix (CP) of size Nmax to obtain
a block of K + Nmax symbols (i.e., one OFDM symbol),
which are subsequently multiplexed and digital to analog
(D/A) converted for transmission. These operations along with
the corresponding FFT and CP removal at the receiver convert
each user’s frequency-selective channel to a set of K parallel
flat-fading subchannels, each with fading coefficient given by
the frequency response of this user’s channel evaluated on the
corresponding
subcarrier. Consider the K × M matrix H̃ :=
√
(1/ K)FK H, whose mth column comprises the frequency
response of user m’s channel.
With the multi-user channel matrix H̃ acquired (via training
symbols), the receiver has available a noise-normalized chan2
nel power gain matrix G, where [G]k,m := |[H̃]k,m |2 /σk,m
,
2 Orthogonal access schemes other than TDMA are also possible. But as we
will see later, the one chosen is not particulary important because the optimal
choice will typically correspond to no sharing; i.e., each subcarrier will be
owned by a single user.
2
with σk,m
denoting the known variance of the zero-mean
additive white Gaussian noise (AWGN) at the receiver. We
will use gk,m := [G]k,m to denote the instantaneous noisenormalized channel power gain for the kth subchannel of the
mth user. Likewise, letting Ḡ := EG [G], its generic entry
ḡk,m := [Ḡ]k,m shall denote the average gain of the (k, m)
subcarrier-user pair. Having (practically perfect) knowledge of
each G realization, the access point (AP) allocates subcarriers
to users after assigning entries of G to appropriate quantization
regions they fall into. Using the indices of these regions, the
receiver feeds back the codeword c = c(G) for the users to
adapt their transmission modes (power, rate and subcarriers)
from a finite set of mode triplets.
Our work relies on the following assumptions:
(as1) Different user channels are uncorrelated; i.e., the
columns of G are uncorrelated.
(as2) Each user’s subchannels are allowed to be correlated,
and complex Gaussian distributed; i.e., gk,m obeys an exponential PDF fgk,m (gk,m )= (1/ḡk,m )exp(−gk,m /ḡk,m ).
(as3) Subchannel states (regions) remain invariant over at
least two consecutive OFDM symbols.
(as4) The feedback channel is error-free and incurs negligible
delay.
(as5) Symbols are drawn from quadrature amplitude modulation (QAM) constellations so that the resulting instantaneous
BER can be approximated as (κ1 = 0.2, κ2 = 1.5)
µ
¶
−pk,m κ2 gk,m
²(pk,m , gk,m , rk,m ) ' κ1 exp
.
(1)
(2rk,m − 1)
(as6) A realization of each gk,m gain falls into one of Lk,m
Lk,m
disjoint regions {Rk,m|l }l=1
.
Since users are sufficiently separated in space (as1) is generally true; (as2) corresponds to fading amplitudes adhering to
the commonly encountered Rayleigh model but generalizations
are possible; (as3) allows each subchannel to vary from one
OFDM symbol to the next so long as the quantization region it
falls into remains invariant; error-free feedback under (as4) is
guaranteed with sufficiently strong error control codes (especially since data rates in the feedback link are typically low);
the accuracy of (as5) is widely accepted; see e.g., [4]; and
(as6) represents a practical and low complexity quantization.
The ultimate goal in this paper is twofold: (G1) design
a channel quantizer to obtain c; and (G2) given c, find
appropriate allocation matrices P, R, and W. We want to
Lk,m
design P, R, W, and {Rk,m|l }l=1
∀k, m, so that the
average power P̄ is minimized under prescribed average rate
r̄0 := [r̄0,1 , . . . , r̄0,M ]T and average bit error rate (BER)
²̄0 := [²̄0,1 , . . . , ²̄0,M ]T constraints across users.
III. Q UANTIZER AND T RANSMISSION M ODE D ESIGN
A. Problem Formulation
Given (as6), let Rk,m|l := {G : gk,m ∈ Rk,m|l } denote the
set of matrices G for which gk,m belongs to the region Rk,m|l .
Furthermore, let pk,m|l and rk,m|l denote3 respectively, the
3 The subscript l here will be also written explicitly as l(G) in places that
this dependence must be emphasized.
instantaneous power and rate loadings of user m on subcarrier
k given that G ∈ Rk,m|l . Recall that wk,m (G) ≤ 1, and
thus the expected power and bit loadings for the G realization
over the time between successive feedback updates will be
pk,m|l wk,m (G) and rk,m|l wk,m (G), respectively.
Our goal is to minimize the average transmit-power
EG [pk,m|l(G) wk,m (G)] over all subcarriers and users while
satisfying average rate and BER requirements. Specifically,
we want the average rate
PKof any user (say the mth) across all
subcarriers to satisfy k=1 EG [rk,m|l(G) wk,m (G)] ≥ [r̄0 ]m .
To enforce the average BER requirement we will have the
instantaneous BER stay always below a pre-specified BER4 .
Then if ²−1
P denotes the inverse function involved when solving
min
:= min{gk,m | gk,m ∈ Rk,m|l }
(1) w.r.t. pk,m and gk,m|l
represent the worst channel gain, the power loading
min
pk,m|l(G) := ²−1
P (rk,m|l(G) , gk,m|l(G) , [²̄0 ]m )
(2)
will automatically fulfill the BER requirement.
Analytically, the constrained optimization problem we wish
to solve is:

minR(G)≥0,W(G)≥0 P̄ , where P̄ :=



 PK PM E [²−1 (r
min


k,m|l(G) , gk,m|l(G) , [²̄0 ]k )wk,m (G)]
k=1
m=1 G P



subject to :
PK

C1. − k=1 EG [rk,m|l(G) wk,m (G)] + [r̄0 ]m ≤ 0, ∀m,


PM


 C2.

m=1 wk,m (G) − 1 ≤ 0, ∀k, G,


C3. − rk,m|l ≤ 0, ∀k, m, l, C4. − wk,m (G) ≤ 0, ∀k, m, G,
(3)
where through the second constraint we enforce the total
utilization of any subcarrier by all users not to exceed one,
per G realization.
Although the optimization problem in (3) is not
convex, defining the auxiliary variable
r̄k,m|l
=
R
rk,m|l w̄k,m|l , with w̄k,m|l
:=
w
(G)
k,m
G∈Rk,m|l
fG (G)dG we can render it convex. Since w̄k,m|l 6= 0
the resultant convex optimization problem using globally
convergent interior point algorithms [1]. The final formulation
now enjoys both reduced complexity and global convergence
in polynomial time thanks to convexity.
The objective in (3) is to minimize the average power
over all possible channel realizations. However, the constraints
involve different forms of CSI: while C2 and C4 needs to
be satisfied per channel realization (and thus will change
depending on G); C1 is an average requirement and C3
is unique per region, and therefore their values do not depend of the concrete channel realization G. In the following
subsection, we will derive the Karush-Kuhn-Tucker (KKT)
conditions associated with (3). These will lead us not only
to the expressions determining the optimal loading variables
but will also provide valuable insights about the structure of
the power-efficient resource allocation policies. At this point,
4 Although this relaxation may lead to a suboptimal solution, it leads to a
less complex optimization problem that turns out to be convex. Numerical
results will show that solution of the relaxed problem attains comparable
performance to that reached by the optimum solution.
a remark is due on another aspect related to power efficiency
in OFDMA.
Remark 1: Although
the
peak-to-average-power-ratio
(PAPR) plays an important role in power (battery)
consumption of OFDM systems, in (3) we did not impose
PAPR constraints. The underlying reason is that available
digital predistortion schemes can be applied to the users’
OFDM symbols to meet such constraints, see e.g., [8].
B. Optimal Policies
p
r
r
w
Let βm
, βkw , αk,m|l
, αk,m|l
, αk,m
denote the positive
Lagrange multipliers associated with C1-C4, respectively.
Specifically, upon defining κ3,m := κ−1
2 ln(κ1 /[²̄0 ]m ) and
setting the derivative of the dual Lagrangian function in (3)
with respect to (w.r.t.) the auxiliary variable r̄k,m|l equal to
zero, yields after tedious but straightforward manipulations the
following KKT condition expressed in terms of the original
variable5
à r∗
!
min
r∗
)gk,m|l(G)
(βm + αk,m|l
∗
rk,m|l(G) = log2
.
(4)
ln(2)κ3,m
p∗
∗
Because KKT conditions for C3 dictate rk,m|l
αk,m|l
= 0 [1],
∗
r∗
for rk,m|l > 0, we need αk,m|l = 0 in (4) and therefore the
min
r∗ −1
condition gk,m|l(G)
> ln(2)κ3,m (βm
) has to be satisfied in
order for the optimum rate loading in the region to be nonzero.
Intuitively, this condition eliminates from the optimum allocation set the regions Rk,m|l with very poor channel conditions
r∗
r∗
(αk,m|l
> 0), while for the remaining regions αk,m|l
= 0.
Furthermore, it is worth noting that due to the logarithmic
expression of ²−1
P [c.f. (1)], the rate loading is reminiscent of
the classical capacity water-filling solution.
Before analyzing the optimality condition for wk,m (G), let
us define the power cost of user m utilizing subcarrier k as
∗
(2rk,m|l(G) − 1)κ3,m
r+∗ ∗
Pk,m (G) :=
− βm
rk,m|l(G) .
min
gk,m|l(G)
(5)
Supposing that Rk,m|l is active, using (5), and differentiating the Lagrangian of (3) w.r.t. wk,m (G), we find at the
optimum
w∗
Pk,m (G)fG (G)+βkw∗ (G)−αk,m
(G) = 0, ∀G, ∀m ∈ [1, M ].
(6)
It is useful to check three things: (i) the LHS(6) does not
∗
(G) but only through the associated
depend explicitly on wk,m
w∗
w∗
multipliers βk (G) and αk,m
(G); (ii) the multiplier βkw∗ (G)
is common ∀m; and (iii) for the same subcarrier k and a given
realization G, the power cost Pk,m (G) is fixed and in general
different for each user m. Furthermore, for each k, the KKT
condition corresponding to C4 also dictates
∗
w∗
wk,m
(G)αk,m
(G) = 0, ∀G, ∀m ∈ [1, M ].
∗
{wk,m
(G)}M
we
m=1 ,
∗
user mk , and wk,m
(G)
(7)
Proposition 1: In the set
have
∗
wk,m
(G)
=
1
for
a
unique
= 0
k
for m 6= mk . Moreover, the user mk assigned to
5 Henceforth,
x∗ will denote the optimal value of x.
utilize the kth subchannel is the one that satisfies: (S1.2) Check C1 (3) ∀m; if |LHS(C1)| < δ[r̄0 ]m ∀m, then
mk = arg minm {Pk,m (G)}M
go to (S1.3); otherwise, increase [β r ]m for the users m
m=1 .
∗
w∗
Proof: Assume that wk,mk (G) > 0, then αk,mk (G) = 0
whose LHS(C1) > 0; decrease [β r ]m for the users m
w∗
and (6) implies that βk (G) = −Pk,mk (G)fG (G). If also
whose LHS(C1) < 0; and go to (S1.1).
∗
r∗
∗
0
w∗
, W∗ are obtained, use (9) to calculate the
wk,m
6= mk then αk,m
0 (G) > 0 with mk
0 (G) = 0, (S1.3) Once R , β
k
k
0
finally allocated power.
and (6) for mk becomes Pk,mk (G) = Pk,m0k (G) ∀G;
∗
which is not true and thus wk,m
0 (G) > 0 is not either
k
∗
0
(uniqueness). If now wk,m
The convexity of (3) enables efficient methods to update
0 (G) = 1 for a mk 6= mk , then
k
w∗
βk (G) = −Pk,m0k (G)fG (G) and using (6) for mk we β r [1]. For example, we can set the initial value of β r
w∗
can write αk,m
(G) = [Pk,mk (G) − Pk,m0k (G)]fG (G) ≥ 0, equal to any small number and update each component [β r ]m
k
separately ∀m by fixing [β r ]m0 , ∀m0 6= m from the previous
which is not true if Pk,mk (G) < Pk,m0k (G) (minimum).
iteration. The adaptation of each [β r ]m is then performed
Notice that if for a given G the minimum value of using line search until the rate constraint for the mth user
{Pk,m (G)}M
m=1 is attained by more than one user, any arbiis tightly satisfied. This simple algorithm has guaranteed contrary time sharing of the subcarrier k among them is optimum; vergence and facilitates computation distributed across users.
or we can simply pick one of them at random without altering
the power cost. Finally, if there exists a realization G such that C. Codeword Structure
Given the quantizer design, we developed so far resource
Pk,m (G) > 0 ∀m, then since βkw∗ (G) ≥ 0 it follows from
w∗
allocation
policies to assign rate, power and subcarriers across
(6) that αk,m (G) 6= 0, ∀m; and the optimal solution will not
users.
Once
the quantizer and resource allocation strategy are
allocate this subcarrier to any user. Therefore, introducing a
∗
designed,
the
AP quantizes each fading state and feeds back a
fictitious Pk,0 (G) = 0 ∀k and G, we can express wk,m (G)
codeword
that
identifies the user-subcarrier assignment and the
in compact form using the indicator function as
region
index
each
subchannel falls into per fading realization
∗
(8) G. Based on this form of Q-CSIT, each user node is informed
wk,m
(G) = I{m=arg minm0 {Pk,m0 (G)}M0 } .
m =0
Since so far we have obtained conditions that the optimal about its own subset of subcarriers (if any) and relies on the
∗
∗
rk,m|l
and wk,m
(G) should satisfy, what remains is a condition region indices to retrieve the corresponding power and rate
∗
for pk,m|l . Although in principle p∗k,m|l can be calculated by levels from a lookup table. The following proposition describes
∗
∗
substituting rk,m|l
and wk,m
(G) into (2), recall that to render the construction of this codeword.
design and the optimal
the optimization problem tractable we had enforced via (2) an Proposition 2: Given the ∗ quantizer
∗
∗
allocation
parameters
(P
,
R
,
W
(β r∗ )) returned by Alinstantaneous BER constraint that is a more restrictive than the
1, the AP broadcasts to the users the codeword
original average BER constraint. Capitalizing on this fact, a gorithm
∗
c
(G)
=
[c∗1 (G), . . . , c∗K (G)] specifying the optimal resource
different approach to obtain the power per region (denoted by
+∗
allocation for the current fading state, where c∗k (G) =
∗
∗
p+∗
k,m|l ), given rk,m|l and wk,m (G), consists of finding pk,m|l
[m∗k (G), lk∗ (G)]T is determined ∀k as:
to satisfy
Lk,m M
1) m∗k (G) = arg min{Pk,m (G, P∗ , R∗ , β r∗ , {β ²∗ }l=1
)}m=1
+∗
∗
∗
m
EG∈Rk,m|l [²(pk,m|l , gk,m , rk,m|l )wk,m (G)]
(pick randomly any user m∗k when multiple minima
∗
(G)].
= [²̄0 ]m EG∈Rk,m|l [wk,m
(9)
occur); and
2) lk∗ (G) = { l | G ∈ Rk,m∗k (G),l , l = 1, . . . , Lk }.
Using this approach we, guarantee that the expected BER
The structure of c∗ (G) in Proposition 2 encodes inaveraged over all channel realizations inside the region Rk,m|l
formation pertinent to each subcarrier (namely, its region
satisfies the pre-specified requirement [²̄0 ]m . Since p+∗
k,m|l <
and assigned user) which is more efficient in terms of
p∗k,m|l , we will rely on (9) to calculate the final power
the number of feedback bits relative to encoding each
loading. Note also that, although (9) involves an integration
user’s individual
+∗
´myielding a codeword length of
³P information,
and pk,m|l can not be found in closed-form, the monotonicity lPK
M
(L
)
bits.
log
k,m
2
m=1
k=1
of the exponential inside the instantaneous BER in (1) allows
+∗
We conclude this section by emphasizing that R (and thus
obtaining pk,m|l through line search (using e.g., the bisection
P) in (3) are involved only in average quantities. Hence,
method).
∗
∗
The following algorithm summarizes the main steps for R and P are computed off-line and only the subcarrier∗
∗
∗
user assignment (involved in instantaneous constraints) and the
finding the optimal allocation of R , W and P .
indexing of the corresponding entries of these matrices need to
be fed back on-line. Thus, almost all the complexity is carried
Algorithm 1: Resource Allocation
out off-line (Algorithm 1), while only a light computation
(S1.0) Select a small positive number δ, and initialize β r using (Proposition 2) has to be carried out on-line.
an arbitrary non-negative vector.
IV. Q UANTIZER D ESIGN
(S1.1) For each (k, m, l) triplet per iteration:
r
In the previous section we addressed our objective (G2) to
(S2.1.1) Use [β r ]m to determine rk,m|l via (4)(αk,m|l
= 0).
derive optimum subcarrier, rate, and power allocation policies
(S2.1.2) Use [β r ]m to determine wk,m (G) via (8).
Pr(w k,m = 1|g k,m )
1
0.5
0
1
2
3
4
5
1
2
3
4
5
1
fgk,m (g k,m )
A. Equally Probable Region Quantizer
To design the quantizer, we first solve the optimal resource
allocation problem supposing CSI is available without quantization (i.e., Lk,m = ∞), and subsequently calculate τk,m|l to
satisfy
R τk,m|l+1
Pr(wk,m = 1, gk,m )dgk,m
τk,m|l
R τk,m|L
k,m
= τk,m|1
Pr(wk,m = 1, gk,m )dgk,m /Lk,m , (10)
0.5
0
Pr(w k,m = 1, g k,m )
assuming the quantization regions, Rk,m|l , are given. In this
section, we will address (G1) by deriving a quantizer that
enforces equally probable quantization regions.
0.06
0.04
0.02
0
1
2
3
4
Normalized Subcarrier Power Gain g k,m / g k,m
5
with τk,m|1 = 0 and τk,m|Lk,m +1 = ∞. If the joint
probabilities can be computed, solving (10) yields thresholds
Fig. 2. Equally Probable Regions Quantizer (K = 64, M = 6, L = 5).
k.m
{τk,m|l }L
l=1 per subcarrier k and user m that divide the joint
probability Pr(wk,m = 1, gk,m ) into regions of equal area;
hence the term equally probable region quantizer. Intuitively
∞
Excluding the case Pk,m
(gk,m ) > 0, which amounts to
speaking, this quantizer design tries to maximize the entropy w
=
0
and
solving
(13)
w.r.t.
γk,µ yields
k,m
in the feedback link.6
κ3,µ ln(2)/βµr∞∗
To evaluate Pr(wk,m = 1, gk,m ) needed in (10), we apply
"
γk,µ (gk,m ) = −
r∞∗ # ,
κ
³
´ βm
Bayes’ rule to re-write it as Pr(wk,m = 1|gk,m )fgk,m (gk,m )
r∞∗
− g 3,m
r∞∗
e ln(2)κ3,m βµ
βµ
W
−1
k,m
f
−2
e
r∞∗
and recall that fgk,i (gk,m ) is known per (as2). To calculate
gk,m βm
Pr(wk,m = 1|gk,m ), we will need to first solve the optimal
(14)
resource allocation problem assuming no quantization. Clearly, where f W [x] = y is the real-valued Lambert’s f W function
∞
as Lk,m → ∞, we have gmin → g in (5). Letting Pk,m
and which solves the equation yey = x for −1 ≤ y ≤ 0 and
r∞∗
βm denote, respectively, the cost indicator and the Lagrange −1/e ≤ x ≤ 0 [3].
multiplier when Lk,m → ∞, we can write
Using (12)-(14), we express Pr(wk,m = 1, gk,m ) in (10) as
¶
µ
r∞∗
r∞∗
βm
κ3,m
gk,m βm
∞
r∞∗
e−gk,m /ḡk,m
Pk,m (gk,m ) =
−
− βm log2
. (11)
Pr(wk,m = 1, gk,m ) = I
ln(2)κ3,µ
ḡk,m
ln(2)
gk,m
(κ3,m ln(2))
{gk,m > β r∞∗
}
m
¡
¢
QM
Because equation (8) establishes that wk,m (G) =
× µ=1,µ6=m 1 − e−γk,µ (gk,m )/ḡk,µ .
(15)
∞
∞
(gk,m ) < Pk,0
= 0,
I{m=arg minm0 {P ∞ 0 (G)}M0 } , if Pk,m
m =0
k,m
r∞∗ M
r∞∗
then gk,m > ln(2)κ3,m /βm
is a necessary condition for the Since (15) depends on {βµ }µ=1 , we need to solve the
problem as Lk,m → ∞. The thresholds
user m to be active. Taking also into account that channels of optimal Lallocation
k,m
{τk,m|l }l=2 are obtained by solving (10) using a line search.
different users are uncorrelated [cf. (as1)], we can write
Notice that we can also take advantage of the condition
∞
r∞∗
Pr(wk,m = 1|gk,m ) = I
ln(2)κ3,m
Pk,m
(gk,m ) < 0, by setting τk,m,2 ≥ κ3,m ln(2)/βm
.
{gk,m >
}
r∗
λm
An
example
illustrating
this
quantizer
is
given
in
Fig.
QM
∞
∞
< Pk,µ
|gk,m ).
× µ=1,µ6=m Pr(Pk,m
(12) 2 which depicts Pr(w
k,m = 1|gk,m ), fgk,m (gk,m ) and
Interestingly, for the active ³users it holds that Pr(wk,m = 1, gk,m ) versus gk,m /ḡk,m for M = 6, Lk,m = 5,
r∞∗
βm
κ3,m
∞
r∞∗
equal average subcarrier gains, and equal rate constraints. The
Pk,m
(gk,m ) = I
− gk,m
−βm
ln(2)κ3,m
ln(2)
{gk,m > β r∞∗ }
first subplot in this figure, Pr(wk,m = 1|gk,m ), reveals that the
m
³
´´
∞
r∞∗
∂Pk,m (gk,m )
gk,m βm
better the channel the more likely the corresponding user is
log2 (ln(2)κ
,
and
therefore
=
∂gk,m
3,m ) ³
´
r∞∗
to be selected. Coupling this observation with the exponential
βm
κ3,m
κ3,m
I
− ln(2) gk,m ≤ 0, which implies
ln(2)κ3,m
{gk,m > β r∞∗
} gk,m
behavior of fgk,m (gk,m ) in the second subplot, the bell-ring
m
∞
(gk,m ) is monotonically decreasing. Therefore, we characteristic of the joint PDF, Pr(wk,m = 1, gk,m ), results
that Pk,m
can find unique channel gains γk,µ , ∀µ 6= m, such that
naturally in the third subplot (after pair-wise multiplication of
the functions in the first two subplots), where the quantization
∞
∞
Pk,m
(gk,m ) = Pk,µ
(γk,µ ).
(13)
thresholds (and regions) resulting from (10) are also identified.
∞
∞
It is then clear that Pk,m
(gk,m ) ≤ Pk,µ
(gk,µ ) if gk,µ ≤ γk,µ , Remark 2: With the method presented in this section, calcu∞
∞
and Pk,m
(gk,m ) > Pk,µ
(gk,µ ) if gk,µ > γk,µ . And conse- lation of τk,m|l based on (10) has to be executed only once
∞
∞
quently, Pr(Pk,m < Pk,µ |gk,m ) = Pr(gk,µ < γk,µ |gk,m ).
(to solve (G1)). With the thresholds available, the resource
allocation can be easily obtained through Algorithm 1 (to
6 The simplicity and efficiency of using an equally probable quantizer were
solve (G2)). Numerical results in the next section will show
first pointed out in [7] for the basic case of a single user and subcarrier. We
here largely expand the scope of this quantizer and apply it to the challenging that this low complexity non-iterative design exhibits power
multiple-user/multiple-subcarrier scenario.
consumption similar to that of the P-CSIT solution.
TABLE I
T OTAL AVERAGE TRANSMIT- POWER ( IN dBW ) FOR P-CSIT, Q-CSIT-EP
AND Q-CSIT-UN SCHEMES .
CASE
Reference Case
²̄0 = 10−4
r̄ = [30, 30, 30]T
K = 128
M =6
SNR = 0
SNR = 10
Q-CSIT-UN
36.1
37.2
29.0
32.2
43.5
39.2
29.5
Q-CSIT-EP
28.6
30.5
23.8
27.0
36.8
31.7
21.7
P-CSIT
28.2
30.1
23.2
26.3
36.2
31.2
21.3
TABLE II
T OTAL AVERAGE TRANSMIT- POWER ( IN dBW ) AS Lk,m
Lk,m
P
2
27.9
3
25.8
4
25.1
5
24.7
VARIES .
∞ (P-CSIT)
24.2
V. N UMERICAL E XAMPLES
To numerically test our power-efficient designs, we consider
an adaptive OFDMA system with M = 3 users, K = 64
subcarriers, noise power per user and subcarrier at 0 dBW ,
Lk,m = 5 regions (i.e., 4 active regions) per subcarrier, and
[²̄0 ]m = ²̄0 = 10−3 ∀m. The average signal-to-noise-ratio
(SNR) considered was 3 dB; and three uncorrelated Rayleigh
taps were simulated per user.
Test Case 1 (Comparison of allocation schemes): For comparison purposes we will contrast the power consumption of
our proposed quantization and allocation scheme (denoted by
Q-CSIT-EP) with the power consumption of the allocation
based on P-CSIT (that represents a lower bound) as well as
with the power consumption of a heuristic scheme based on
Q-CSIT with uniform quantization and subcarrier allocation
but optimum rate and power loading (denoted by Q-CSITUN). Numerical results assessing the performance of P-CSIT,
Q-CSIT-EP and Q-CSIT-UN schemes over a wide range
of parameter values are summarized in Table I, where the
reference case is K = 64, M = 3, r̄0 = [60, 60, 60]T ,
SNR = 3 dB; and the “column case” entails only a single
variation w.r.t. the reference case. The striking observation
here is the almost equivalent performance of Q-CSIT-EP and
P-CSIT schemes. These results certify the usefulness of the
proposed quantizer based on equally probable regions and
validates the optimality of Algorithm 1. Moreover, the power
loss of at least 6 dB with respect to Q-CSIT-UN shows the
important role of subcarrier allocation and quantization in
terms of minimizing the transmit-power.
Test Case 2 (Number of quantization regions): Table II
lists the average transmit-power when varying the number of
regions per subcarrier for r̄0 = [20, 40, 60]T . Notice that
usually the number of active regions is equal to Lk,m − 1,
since the optimum solution yields one outage region (for the
cases when the channel gain is very poor). Simulation results
demonstrate that our proposed equally probable quantization
and Algorithm 1 used together lead to a power loss no greater
than 3-5 dB with respect to the P-CSIT case (Lk,m = ∞).
Moreover, the resulting power gap shrinks as the number of
regions increases reaching a power loss of approximately only
0.5 dB in the case of four active regions.
VI. C ONCLUDING S UMMARY AND F UTURE R ESEARCH
Based on Q-CSIT, we devised a power efficient OFDMA
scheme under prescribed individual average rate and BER constraints. In this setup, an access point quantizes the subcarrier
gains and feeds back to the users a codeword conveying the
optimum power, rate, and subcarrier policy. The resulting nearoptimal transceivers are attractive because they only incur a
power loss as small as 1 dB relative to the benchmark design
based on perfect CSIT which requires feedback information
which may be unrealistic in wireless systems.
Our novel design separates resource allocation from channel
quantization. The first was obtained as the solution of a
convex constrained optimization problem, while the second
was designed to ensure equally probable regions per user
and subcarrier. We ended up with a lightweight resource
allocation protocol where both rate and power are available at
the transmitter through a lookup table and only the subcarrier
assignment needs be determined on-line.
To build on the presented framework, future directions will
exploit the possible correlation across subcarriers to group
subcarriers and then index each group.7
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7 The views and conclusions contained in this document are those of the
authors and should not be interpreted as representing the official policies,
either expressed or implied, of the Army Research Laboratory or the U. S.
Government.